1. Patterns – Learning Outcomes
Use tables to represent a repeating-pattern situation.
Generalise and explain patterns and relationships in words and numbers.
Use tables, diagrams, and graphs for representing and analysing linear, quadratic, and exponential patterns and relations.
Find the underlying formula written in words from which the data are derived for linear and HL: quadratic relationships.
Write arithmetic expressions for particular terms in a sequence.

2. Explain Patterns in Words and Numbers
Describe the pattern below:
Can you describe just which ones are blue? What about which ones are red?
If the pattern continues, can you decide the colour of the 15th box?
What colour is the 20th box?
What colour is the 100th box?

3. Explain Patterns in Words and Numbers
When the length of a pattern is two, it’s easy to make predictions: divide the pattern into odds and evens.
For longer patterns, we do the same, but divide the pattern into more groups.
e.g. how often does the pattern shown below repeat?

4. Use Tables to Represent Patterns
A table may help to make predictions.
Number 1 2 3 4 5 6 7 8 9 10 11 12
Instead of grouping them as odds and evens, we group them into three groups – stars, circles, and triangles.
What do the numbers in the triangle group have in common?
{3, 6, 9, 12…}
They’re all multiples of three!

5. Use Tables to Represent Patterns
Each number can be sorted into their group by their remainder when divided by three:
Number 1 2 3 4 5 6 7 8 9 10 11 12
Remainder
All stars have remainder 1.
All circles have remainder 2.
All triangles have no remainder.
How do we calculate remainders?

6. Use Tables to Represent Patterns
To find the remainder:
Divide the number by 3.
Make sure the result is in mixed fraction mode: press SHIFT, then 𝑆⇔𝐷 if not.
Make sure the denominator matches the original divisor.
The numerator is the remainder.
e.g. 65
65÷3=21 2 3
⇒Remainder =2 (circle)
If they don’t match, it will require further work – we’ll get to that later.

7. Use Tables to Represent Patterns
what shape is the 4th symbol?
what shape is the 5th symbol?
what shape is the 6th symbol?
what shape is the 21st symbol?
what shape is the 22nd symbol?
what shape is the 23rd symbol?
what shape is the 43rd symbol?
what shape is the 53rd symbol?

8. Use Tables to Represent Patterns
e.g. How often does the pattern shown repeat? A B C D E F
Draw a table for the first 12 letters in the pattern.
Use remainders to divide the letters into groups (e.g. all As have remainder 1).
Calculating remainders is tricky here.
e.g. What letter is in position 10?
Usually, 10÷6= would give remainder 2.
The denominator and divisor much match however:
⇒ 2 3 = 4 6 , so the remainder is 4, which is a D.

9. Use Tables to Represent PatternsC D E F
Number 1 2 3 4 5 6 7 8 9 10 11 12
Remainder
What letter is in position 14?
What letter is in position 15?
What letter is in position 16?
What letter is in position 26?
What letter is in position 28?

10. Explain Patterns in Words and Numbers
Consider the sequence:
3, 9, 15, 21, 27, …
Imagine each number in a box in sequence: 3 9 15 21 27
Box #1
Box #2
Box #3
Box #4
Box #5
Draw out the next three boxes.
What number is in box #10?
What box contains the number 75?
What number is in box n?

11. Explain Patterns in Words and Numbers3 9 15 21 27
Box #1
Box #2
Box #3
Box #4
Box #5
The number in box #1 is called the start term, a.
The difference between boxes is called the common difference, d.
The number in box #n is called the nth term, Tn. (e.g. box #5 contains the 5th term, T5, box #25 contains the 25th term, T25).
Only linear patterns have a common difference!

12. Explain Patterns in Words and Numbers
Write the first five terms of the following sequences:
start term = 2, common difference = 5
start term = 5, common difference = 2
start term = 18, common difference = -2
Describe in words how to find the nth term of each of these sequences.
Write an expression in terms of n to describe the nth term of these sequences.

13. Explain Patterns in Words and Numbers
Write the first four terms of each of the following sequences:
𝑇 𝑛 =5𝑛+1
𝑇 𝑛 =2𝑛+4
𝑇 𝑛 =9 4−2𝑛
Write down the start term and the common difference for each of the sequences above.

14. Use Tables to Represent Linear Patterns
Disc-shaped tiles are placed to form a pattern as shown:
Draw the next two stages of the pattern.
Draw a table showing how many tiles are in each stage of the pattern.
Write down the general term, 𝑇 𝑛 , for the number of tiles in stage 𝑛 of the pattern.
𝑇 𝑛 =𝑎+ 𝑛−1 𝑑

15. Use Tables to Represent Linear Patterns
2015 FL Q14
A pattern is made using white tiles and shaded tiles. Here are the first two terms in the pattern.
Draw Term 3 of the pattern in the grid above.

16. Use Tables to Represent Linear Patterns
2015 FL Q14
[continued]
Fill in the fraction boxes below to show what fraction of each term is shaded.
Draw a table showing the number of white tiles and the number of shaded tiles in the first five terms of the pattern.
Ciarán draws another term of the pattern. It has 14 white tiles. How many shaded tiles should it have?

17. Use Tables to Represent Linear Patterns
2014 OL P1 Q11
The first three stages of a pattern are shown below. Each stage is made up of a certain number of shaded discs and a certain number of white discs.
Shade in appropriate discs below to show the 4th stage of the pattern.

18. Use Tables to Represent Linear Patterns
2014 OL P1 Q11
[continued]
Complete the table below to show how the pattern continues.

19. Use Tables to Represent Linear Patterns
2014 OL P1 Q11
[continued]
In a particular stage of the pattern, there are 21 white discs. How many shaded discs are there in this stage of the pattern?
Write down the relation between the number of shaded discs and the number of white discs in each stage of the pattern. State clearly the meaning of any letters you use.
𝑇 𝑛 =𝑎+ 𝑛−1 𝑑

20. Use Tables to Represent Linear Patterns
2016 HL P1 Q8
John makes a sequence where each stage is made up of a certain number of Xs arranged in a pattern. The first three stages of John’s sequence are shown below. The sequence starts at stage 0.
Draw the next stage of John’s sequence.

21. Use Tables to Represent Linear Patterns
2016 HL P1 Q8
[continued]
Using a table, a graph, or otherwise, write a formula to express 𝑁 in terms of 𝑆, where 𝑁 is the number of Xs in stage 𝑆 of John’s sequence.
There are exactly 130 Xs in stage 𝑘 of John’s sequence. Find the value of 𝑘.

22. Use Tables to Represent Linear Patterns
2016 HL P1 Q8
[continued]
Yoko is also making a sequence, with each stage made up of a number of Xs arranged in a pattern. In Yoko’s sequence, the relationship between 𝑁 and 𝑆 is given by the formula: 𝑁=1+2𝑠, where 𝑁 is the number of Xs in stage 𝑆 of the sequence (starting at stage 0).
Draw one possible example of the first three stages of Yoko’s sequence in the table below:
𝑝 represents the number of Xs in stage 𝑦 of Yoko’s sequence. Write down the number of Xs in stage 𝑦+3 of Yoko’s sequence. Give your answer in terms of 𝑝.

23. Use Tables to Represent Quadratic Patterns
Linear patterns have a common difference, d.
Quadratic patterns have a common second difference.
e.g. 3, 6, 11, 18, 27…
Term 3 6 11 18 27
1st Diff +3 +5 +7 +9
2nd Diff +2

24. Use Tables to Represent Quadratic Patterns
Confirm that the following sequences are quadratic and write down the next two terms:
1, 4, 9, 16, 25…
3, 6, 11, 18, 27…
0, 3, 8, 15, 24…
2, 8, 18, 32, 50…
-3, 8, 23, 42, 65…
9, 28, 57, 96, 145…
3, 12, 27, 48, 75…
16, 7, 2, -2, -4…

25. Use Tables to Represent Quadratic Patterns
The term rules for quadratic sequences are in the form 𝑎 𝑛 2 +𝑏𝑛+𝑐. Write down the first four terms of each of the following sequences:
𝑇 𝑛 = 𝑛 2 +2𝑛+1
𝑇 𝑛 = 𝑛 2 −2𝑛+1
𝑇 𝑛 = 𝑛 2 +3𝑛−6
𝑇 𝑛 =2 𝑛 2 +5𝑛−10
𝑇 𝑛 =3 𝑛 2 −6𝑛+4
𝑇 𝑛 =7 𝑛 2 −7𝑛+7

26. Use Tables to Represent Quadratic Patterns
When given terms in a quadratic sequence, we can generate the term rule 𝑎 𝑛 2 +𝑏𝑛+𝑐 using three facts:
𝑎= 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2
By letting 𝑛=0, we get 𝑐= 𝑇 0 . Although this does not normally exist, we can pretend it does for this purpose.
𝑏 can be found by substitution.

27. Use Tables to Represent Quadratic Patterns
e.g. 3, 6, 11, 18, 27 (from pg 23)
Term 3 6 11 18 27
1st Diff +3 +5 +7 +9
2nd Diff +2
𝑎= 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2 = 2 2 =1
Working backwards, 𝑇 0 =3−1=2⇒𝑐=2
e.g. 𝑇 1 = 𝑏 1 +2=3
⇒1+𝑏+2=3
⇒𝑏=0
⇒ 𝑇 𝑛 = 𝑛 2 +2
Recall that in general,
𝑇 𝑛 =𝑎 𝑛 2 +𝑏𝑛+𝑐 for quadratic sequences

28. Use Tables to Represent Quadratic Patterns
Determine the term rule for each of the following quadratic sequences (you can look up the answers from pg 24 to speed this up):
1, 4, 9, 16, 25…
3, 6, 11, 18, 27…
0, 3, 8, 15, 24…
2, 8, 18, 32, 50…
-3, 8, 23, 42, 65…
9, 28, 57, 96, 145…
3, 12, 27, 48, 75…
16, 7, 2, -2, -4…

29. Use Tables to Represent Exponential Patterns
Exponential sequences do not have a common difference, but instead have a common factor.
e.g. 2, 6, 18, 54, 162…
Term 2 6 18 54
162
1st Diff +4
+12
+36
+108
2nd Diff +8
+24
+72
Neither the first nor second differences are common, so try factors:
Term 2 6 18 54
162
Factor ×3

30. Use Tables to Represent Exponential Patterns
Confirm that the following sequences are exponential and write down the next two terms:
6, 18, 54, 162…
6, 12, 24, 48…
10, 20, 40, 80…
12, 36, 108, 324…
-4, -8, -16, -32…
-9, -27, -81, -243…

31. Use Tables to Represent Exponential Patterns
Write out the first four terms of each of the following exponential sequences:
𝑇 𝑛 = 2 𝑛
𝑇 𝑛 = 3 𝑛
𝑇 𝑛 =4 3 𝑛
𝑇 𝑛 =5 2 𝑛
𝑇 𝑛 =10 2 𝑛
𝑇 𝑛 =7 3 𝑛

32. Use Tables to Represent Patterns
2013 HL P1 Q14
Investigate whether the pattern in the table below is linear, quadratic, or exponential. Explain your conclusion.
Term 1
Term 2
Term 3
Term 4
Term 5
2𝑎−𝑏+2𝑐
8𝑎−2𝑏+2𝑐
18𝑎−3𝑏+2𝑐
32𝑎−4𝑎+2𝑐
50𝑎−5𝑎+2𝑐

33. Use Graphs to Represent Patterns
To graph patterns, let 𝑛 be the 𝑥 axis and 𝑇 𝑛 be the 𝑦 axis.
e.g. linear: 3, 9, 15, 21, 27

34. Use Graphs to Represent Patterns
e.g. quadratic: 3, 6, 11, 18, 27

35. Use Graphs to Represent Patterns
e.g. exponential: 2, 6, 18, 54, 162

36. Use Graphs to Represent Patterns
2012 HL P1 Q7
Lisa is on a particular payment plan called “Plan A” for electricity. She pays a standing charge each month even if no electricity is used. She also pays a rate per unit used. The table shows the cost, including the standing charge, of using different amounts of units, in a month.
Units Used
Plan A Cost (€)
100 38
200 56
300 74
400 92
500
110
600
128
700
146
800
164
Use the data in the table to show that the relationship between the number of units used and the cost is linear.

37. Use Graphs to Represent Patterns
2012 HL P1 Q7
[continued]
Draw a graph to show the relationship between the number of units used and the cost of electricity.
Use your graph to estimate the standing charge.
Write down a different method of finding the standing charge. Find the standing charge using your method.
Write down a formula to represent the relationship between the number of units used and the cost for any given number of units.
The table above does not include VAT. One month Lisa used 650 units. Her total bill for that month, including VAT, was € Find the VAT rate on electricity, correct to one decimal place.

38. Use Graphs to Represent Patterns
2012 HL P1 Q7
[continued]
Lisa is offered a new plan, “Plan B”, where the standing charge is €36 and the rate per unit used is 15.5 cent. Add a column to the table for plan A, showing the costs for plan B.
Which plan do you think Lisa should choose? Give a reason for your answer.
On your diagram for part (b), draw a graph to show the relationship between the number of units used and the cost of electricity for Plan B. Label this graph “Plan B”.
Use your graph to find the number of units for which both plans have the same cost.